OFFSET
0,4
FORMULA
P(n,k) = {p prime | k+1 <= p <= n }
L(n,p,r) = Sum_{i>=0} floor((n-r)/((p-r)*p^i))
A(n,k) = Prod_{p in P(n,k)} p^(Sum_{m=0..k} (-1)^m*L(n,p,m))
F(n,k) = A(n,k)^((-1)^k).
EXAMPLE
For n >= 0
F(n,0) 1, 1, 2, 6, 24, 120, 720, 5040, 40320, ...
F(n,1) 1, 1, 1, 4, 2, 48, 16, 576, 144, 3840, ...
F(n,2) 1, 1, 1, 3, 9, 135, 1215, 2835, 127575, ...
F(n,3) 1, 1, 1, 1, 1, 1, 1, 5, 1, 25, 5, 35, ...
F(n,4) 1, 1, 1, 1, 1, 5, 25, 175, 4375, 4375, ...
F(n,5) 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 49, ...
MAPLE
F := proc(n, k) local L, p, i;
L := proc(n, u, r) local q, s, m; m:=n-r;
q:=u-r; s:=0; do if q>m then break fi;
s:=s+iquo(m, q); q:=q*u od; s end;
mul(p^add((-1)^i*L(n, p, i), i=0..k),
p = select(isprime, [$(k+1)..n]))^((-1)^k) end:
a(n) := n -> F(n, 2);
MATHEMATICA
F[n_, k_] := Module[{L, p, i}, L[n0_, u_, r_] := Module[{q, s, m}, m = n0-r; q = u-r; s = 0; While[True, If[q > m, Break[]]; s = s + Quotient[m, q]; q = q*u]; s]; Product[p^Sum[(-1)^i*L[n, p, i], {i, 0, k}], {p, Select[Range[k+1, n], PrimeQ]}]^((-1)^k)]; a[n_] := F[n, 2]; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Jan 15 2014, translated from Maple *)
PROG
(Sage)
def A163402(n):
def L(n, u, r):
m = n - r; q = u - r
s = 0
while(q <= m):
s += m//q
q *= u
return s
P = filter(is_prime, [3..n])
return mul(p^add((-1)^i*L(n, p, i) for i in (0..2)) for p in P)
print([A163402(n) for n in range(20)]) # Peter Luschny, Mar 13 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Jul 26 2009
STATUS
approved